3.1 Estudio y definición de las tecnologías de avanzada que han resultado
3.1.3 Tratamiento de las lesiones en elementos de terminación
4.1. The heat kernel of perturbed Laplacian operators. In this section, we construct the heat kernel of some second order pseudo-differential operators on a groupoid G⇒M with M compact.
Given a vector bundle E over M, fix anA-connection∇Eon E. Then the pull-back defines a (family of s-fiberwise) connection on the bundle s−1E → s−1(x), x ∈ M, which we shall still denote by ∇E. Also, recall that we fixed a metric on A, hence a Riemannian metric on the fibers s−1(x), x∈M, which we shall still denote bygA. We define the Laplacian by taking the trace of the square of ∇E. More precisely:
Definition 4.1. The Laplacian ∆E ∈ Ψ2µ(G) is the family of operators {∆Ex}x∈M, where ∆Ex := n X i=1 (∇EX i∇ E Xi− ∇ E ∇E XiXi), and Xi is any local orthonormal basis of TGx.
Note that ∆E is elliptic, and its principal symbol does not depend on the chosen connection ∇E.
We consider an operator of the form
(18) ∆E+F+K,
where F ∈ Γ∞(t−1E⊗t−1E), considered as a differential operator of order 0; and
K ∈Ψ−∞µ (G,E). We shall denote the reduced kernel ofK by κ.
Since the restriction of t−1E to each s-fiber Gx is a vector bundle with bounded
geometry, we have the Sobolev norms k · k∞,l defined by Equation (39). For u ∈
Γ∞(t−1E), we define
kukl:= sup
x∈M
{ku|s−1(x)k∞,l}.
Denote byt−1E⊗s−1E0n(0,∞) the pullback oft−1E⊗s−1E0 → Gby the projection
G ×(0,∞)→ G.
Definition 4.2. A (groupoid) Heat kernelof ∆E+F+K is a continuous section
Q∈Γ0(t−1E⊗s−1E0n(0,∞)),
such thatQ(a, t), Q(a−1, t) are smooth when restricted to allGx×(0,∞), and satisfies:
(1) The heat equation
(∂t+ ∆E+F+K)Q(a, t) = 0.
Here, we use the fact that s−1E0|Gx ∼=Gx×E 0
x, and let ∆E+F +K to act
on the t−1E factor of Q(a, t)|Gx ∈ Γ
∞(t−1E⊗s−1E0)∼
= E0x×Γ∞(t−1E) for
each tfixed;
(2) The initial condition lim t→0+Q◦u=u, ∀u∈Γ ∞ c (t −1 E),
where◦ denotes the convolution product.
Let Q be a groupoid heat kernel. Then it is clear that for any x ∈ M, (a, b) ∈ Gx×Gx7→Q(ab−1, t) is a heat kernel of (∆E+F+K)x on the manifold with bounded
geometry Gx. Using the uniqueness of the heat kernel on manifolds with bounded
geometry, it is clear that:
Lemma 4.3. A groupoid heat kernel Q of ∆E+F +K, if it exists, is unique. 4.1.1. The formal solution. Before we start, we need to define some notation.
Recall that there exists r0>0 such that exp∇is a diffeomorphism from the set
Ar0 :={X ∈ A:gA(X, X)< r20}
onto its image. For each x ∈M, we denote the polar coordinates on Ax, the fiber
of A over x, by (r, ϑ). The image ofAr0 under exp
∇ is denoted byB(M, r0). Note that since
d(exp∇(r, ϑ), x) =r,
therefore B(M, r0) = {a ∈ G :d(a,s(a)) < r0}, as expected. The exponential map also defines a local trivialization oft−1E: For eacha= exp∇X∈B(M, r0), E ∈Es(a) where X ∈ As(a), define T(a)(E) ∈ t−1Ea to be the parallel transport of E to a
along the curve exp∇τ X, τ ∈ [0,1]. Hence T is a map from the set {(a, E) : a ∈
B(M, r0), E ∈Es(a)} tot−1E|B(M,r0), and we denote its inverse map byT−1. When restricted to t−1E|Gx for some x∈M, the image of T
−1, lies inE
x. In that case we
shall still denote the restricted map by T−1 :t−1E|GxTB(M,r0)→Ex.
Lastly, we letJ := det(dexp∇)◦(exp∇)−1to be the Jacobian, andV :=d(a,s(a))×
Consider a kernel of the form
q(a, t)Φ(a, t)∈Γ∞(t−1E⊗s−1E0n(0,∞)), where q:B(M, r0)×(0,∞)→Ris the Gaussian function
q(a, t) := (4πt)−n2 e− d(a,s(a))2
4t . A straightforward calculation shows that:
Lemma 4.4. One has
(∂t+ ∆E+F)q(a, t)Φ(a, t) =q(a, t)(∂t+ ∆E+F+t−1∇EV +
LVJ
2tJ )Φ(a, t).
Lemma 4.5. There exists a formal power series
Φ(a, t) = ∞
X
i=1
tiΦi(a), Φi∈Γ∞
satisfying the equation
(19) (∂t+ ∆E+F +t−1∇EV +
LVJ
2tJ )Φ(a, t) = 0.
Proof. Equating coefficients one gets
∇EV(J12Φ0) = 0
∇E
V(J
1
2Φi) +iΦi =−(∂t+ ∆E+F)Φi−1, i= 1,2· · · These are simple ordinary differential equations, with explicit solutions
Φ0(expX) =J−12T(expX) Φi(expX) =−J− 1 2T Z 1 0 J12T−1((∂t+ ∆E+F)Φi−1(expτ X))τi−1dτ.
Fix a cutoff function χsupported on B(M, r0) such that χ= 1 on the smaller set
B(M,r0 2) :={a∈ G :d(a,s(a))≤ r0 2}. Write GN(a, t) :=χ(a)q(a, t) N X i=1 tiΦi(a), t∈(0,∞).
Then one has
(1) For any k, l∈N, there exists a constant Ck,l such that k∂tk((∂t+ ∆E+F)GN)kl≤Ck,ltN− n 2−k− l 2;
(2) For any t0>0, the map
u7→GN(·, t)◦u,0≤t≤t0
is a uniformly bounded family of operators on Γl(t−1E), and for any u ∈
Γl(t−1E),
lim
t→0+kGN◦u−ukl= 0.
Proof. OnB(M,r0
2), from the proof of Lemma 4.5, one has (∂t+ ∆E+F)GN(a) =tNq(a, t)ΦN(a) = (4π)−n2tN− n 2e− d(a,s(a))2 4t ΦN(a).
It is elementary that e−d(a,s4(ta))2 is bounded for any a, t, and ΦN is smooth and hence has bounded derivatives. On G\B(M,r0
2) observe that e
−d(a,s(a))
4t and all its derivatives decay faster than any powers as t→0. That proves (1) in the case l=
k= 0. Other cases follow from a similar argument, with the additional observation that ∂te− y2 t =−t−1(y 2 t )e −yt2 =O(t−1) ∂ye− y2 t =−t− 1 2(y 2 t ) 1 2e− y2 t =O(t− 1 2). To prove (2), write for anya∈ G,
GN ◦u(a) := Z Gs(a) GN(ab−1)u(b)µs(a)(b) = Z s−1(t(a))
GN(c−1)u(ca)µt(a)(c) (using the right invariance of µ)
= Z t(a) (4πt)−n2e d(c−1,t(c))2 4t χ(c−1) XN i=0 tiΦi(c−1) u(ca)µt(a)(c).
By right invariance and symmetry of the distance functiond(·,·), one hasd(c−1,t(c)) = d(c,s(c)). Hence χ(c−1) =χ(c), and ed(c
−1,t(c))2 4t =e
d(c,s(c))2
4t . Therefore the inte- grand is supported on B(M, r0) and the integral can be computed by a change of
variable c= exp∇X, X∈ At(a), gA(X, X)≤r20: Z s−1(t(a)) (4πt)−n2e− d(c−1,t(c))2 4t χ(c−1) XN i=0 tiΦi(c−1) u(ca)µt(a)(c) = Z s−1(t(a)) (4πt)−n2e− d(c,s(c))2 4t χ(c) XN i=0 tiΦi(c−1) u(ca)µt(a)(c) = Z X∈At(a) (4πt)−n2e− gA(X,X) 4t χ(exp∇X) × N X i=0 tiΦi((exp∇X)−1)
u((exp∇X)a) det(dexp∇)(X)dX.
It is clear that the last expression converges to
χ(exp∇0)(
N
X
i=0
tiΦi((exp∇0)−1))u((exp∇0)a)(det(dexp∇)(0)) =u(a),
since (4πt)−n2e− gA(X,X)
4t is just the Gaussian heat kernel on the usual Euclidean space.
4.1.2. From parametrix to heat kernel. In the last section we constructed an approximate solution to the heat kernel. In this section we use the method of Levi parametrix to construct a heat kernel. We turn to operators of the form
∆E+F+K.
For each N > n2, define the sectionsR(nk)∈Γ∞(t−1E⊗s−1E0[0,∞)):
R(1)N := (∂t+ ∆E+F+K)GN R(Nk):= Z t 0 RN(·, t−τ)◦R (k−1) N (·, τ)dτ = Z t 0 Z s−1(a) Rn(ab−1, t−τ)R (k−1) N (b, τ)µs(a)(b)dτ Q(0)N :=GN Q(Nk):= Z t 0 GN(·, t−τ)◦RN(k)(·, τ)dτ, k≥1.
Then one has the estimates
Lemma 4.7. Let N > n+2l. There exists constants C˜l, l∈N such that
kR(k)(·, t)kl≤C˜lC˜0kMk(1 +tN −n+l
Proof. Using the same arguments as in the proof of (2) Lemma 4.6, one hasKGN =
κ(·)◦GN(·, t)→κin thek·kl-norm ast→0. ThereforeKGN is a continuous section
over G ×[0,∞), and its l-partial derivatives extends continuously to t ∈ [0,∞). Combining with (1) of Lemma 4.6, it follows that the integrand is a continuous section on G ×[0, t], so the integral exists (and is finite).
Combining (1) of Lemma 4.6 and the boundedness of K to obtain for eachl,
kR(1)N (·, t)kl =k(∂t+ ∆E+F +K)GN(·, t)kl ≤C˜l(1 +tN−
n+l 2 )
for some ˜Cl>0. ExpandR(k) as a multiple integral:
R(Nk)(a, t) = Z 0≤tk−1≤···≤t1≤t Z b1,b2,···bk−1∈s−1(a) R(1)N (ab−11 , t−t1)R (1) N (b1b−12 , t1−t2)· · · ×R(1)N (bk−1bk−1, tk−2−tk−1)RN(1)(bk−1, tk−1)µ(b1)· · ·µ(bk−1).
Next, consider the domain of integration. Since both GN and κ have compact
supports, R(1)N is compactly supported for each t ≥ 0. In particular, there exists
ρ > 0 such that R(1)N (c1c−12 , t) = 0 for any c1, c2 ∈ G such that s(c1) = s(c2) and
d(c1, c2)≥ρ. Using the bounded geometry property of thes-fibers, we take
M := sup
c∈G
Z
B(a,ρ)
µs(c) <∞.
Then it follows that the volume of the domain of integration is bounded by
Z b1∈B(a,ρ) Z b2∈B(b1,ρ) · · · Z bk−1∈B(bk−2,ρ) µs(a)(b1)· · ·µs(a)(bk−1)≤Mk−1.
By elementary calculation, one also gets
Z
0≤tk≤···≤t1≤t
Finally, one has for anya∈ G, |R(Nk)(a, t)|l≤ Z 0≤tk−1≤···≤t1≤t Z b1,b2,···bk−1∈s−1(a) |R(1)N (ab−11 , t−t1)|l ×|R(1)N (b1b2−1, t1−t2)|0· · · |R(1)N (bk−1b−1k , tk−2−tk−1)|0 × |R(1)N (bk−1, tk−1)|0µs(a)(b1)· · ·µs(a)(bk−1) ≤ Z 0≤tk−1≤···≤t1≤t Z b1∈B(a,ρ) Z b2∈B(b1,ρ) · · · Z bk−1∈B(bk−2,ρ) ˜ ClC˜0k−1(1 +tN −n+l 2 )kµ s(a)(b1)· · ·µs(a)(bk−1) ≤C˜lC˜0kMktk −1 (1 +tN−n+2l)k((k−1)!)−1.
The assertion follows by taking supremum over a∈ G.
Lemma 4.8. Assume that l >1,2N > n+l. (1) There exists constants Cl0 such that
kQ(Nk)(·, t)kl≤Cl0C˜0kMk(1 +tN−n2+l)ktk(k!)−1;
(2) The kernelQ(Nk)(a, t) is continuously differentiable with respect to t and
(∂t+ ∆E+F +K)QN(k)=R(Nk+1)+RN(k).
Proof. Define the section
B(a, t, s) := (GN(·, t−s)◦R(Nk)(·, s))(a), ∀a∈ G, t∈[0,∞), s∈[0, t].
Since GN(·, t−s) is Cl by our construction, by (2) of Lemma 4.6, one has for any
0≤s≤t, kb(·, t, s)kl ≤C0 Z t 0 kR(Nk)(·, s)klds ≤C0C˜0kMk(1 +tN−n+2l)k Z t 0 sk−1((k−1)!)−1ds ≤C0C˜0kMk(1 +tN−n+2l)ktk(k!)−1,
from which (1) follows. To prove (2), one has (∂t+ ∆E+F +K)( Z t 0 B(a, t, s)ds)(a, t) =B(a, t, t) + Z t 0 (∂t+ ∆E+F +K)GN(·, t−s)◦RN(k)(·, s)ds =R(Nk)(a, t) + Z t 0 R(0)(·, t−s)◦R(Nk)(·, s)ds =R(Nk)(a, t) +R(Nk+1)(a, t).
Finally, we can construct the heat kernel
Lemma 4.9. For any l, N with 2N > n+l+ 1, the series
∞
X
k=0
(−1)kQ(Nk)(·, t)
converges to a limit Q(·, t) ∈ Γ0(t−1E⊗s−1E0 ×(0,∞)), independent of N, in the k · kl norm. Furthermore,
(1) The sectionQ is the heat kernel of∂t+ ∆E+F +K;
(2) GN approximatesQ in the sense that
kQ−GNkl =O(t)
as t→0.
Proof. From (1) of Lemma 4.8, one hasQ(Nk)< 21k for sufficient largek. Convergence of the series P∞
k=0(−1)kQ (k)
N follows from the comparison test. Assertion (2) follows
from Q−GN =
P∞
k=1Q (k)
N , and implies the initial condition of (1), i.e.,
lim
t→0+kQ◦u−ukl= 0, since
To show that (∂t+ ∆E+F+K)Q= 0, observe thatk(∂t+ ∆E+F+K)Q(Nk)kl≤2−k
for sufficient large k. Therefore one has
(∂t+ ∆E+F +K) ∞ X k=1 (−1)kQ(Nk)= ∞ X k=1 (−1)k(∂t+ ∆E+F+K)Q(Nk) =R(1)N + ∞ X k=2 (−1)k(RN(k)+R(Nk−1)) = 0.
Notation 4.10. We shall denote the heat kernel of the Laplacian ∆E+F +K, as constructed above, by
e−t(∆E+F+K):=Q(·, t).
Remark 4.11. Alternatively, lete−t(∆E+F) be the heat kernel of ∆E+F constructed using the same method as above. Then a heat kernel of ∆E+F +K is given by
(20) e−t(∆E+F+K) =e−t(∆E+F)+ ∞ X i=1 tiQ˜(i), where ˜ Q(i):= Z 0<τ0<···<τi<1 e−t(∆E+F)(·, τ0t)◦κ◦e−t(∆ E+F) (·, τ1t)◦κ◦· · ·◦κ◦e−t(∆ E+F) (·, τit),
and the integration is over the Lebesgue measure.
As in the case of manifolds with bounded geometry, the heat kernel of Laplacian on groupoids satisfies the following ‘off diagonal’ estimate:
Proposition 4.12. Fixε >0such that for any a∈ G,κ(ab−1) = 0andGN(ab−1, t)
= 0 for any t, wheneverb ∈ Gs(a)\B(a, ε). Let t >0 be fixed. For any λ > 0, there exists C >0 such that
(21) |e−t(∆E+F+K)(a, t)| ≤Ce−λd(a,s(a)), ∀a∈ G, d(a,s(a))>2ε,
Proof. Let I ∈N be such that Iε ≤ d(a,s(a)) ≤ (I + 1)ε. ThenQ(Nk)(a, t) = 0 for any k < I. Therefore one has
|Q(a, t)|eλd(a,s(a)) ≤ ∞ X k=I eλ(I+1)εC00C˜0kMk(1 +tN−n2)ktk(k!)−1 =eλ(I+1)εC 0 0C˜0IMI(1 +tN− n 2)ItI I! × ∞ X k=0 ˜ C0kMk(1 +tN−n2)ktkI! (k+I)! . It is clear that the last expression goes to 0 as I → ∞, so Equation (21) is proved. From Equation (21), one has
Q(a−1, t)≤Ce−λd(a−1,t(a))=Ce−λd(a,s(a)).
It follows that Q(a−1, t) ∈ L1(Gs(a)) because Gs(a) has at most exponential volume
growth.
4.1.3. The heat kernel of the vector representation. We turn to study the heat kernel ofν(∂t+∇E+F+K), whereνis the vector representation. The construction
becomes very simple, once we know the heat kernel of (∂t+∇E+F+K).
Theorem 4.13. IfQis a heat kernel of∂t+∇E+F+K, thenν(Q) is a heat kernel
of ν(∂t+∇E+F +K) in the sense that
ν(∂t+∇E+F +K)ν(Q)f = 0, ∀t >0
(22)
lim
t→0+kν(Q)f −fk= 0
for any f ∈Γ∞(E).
Proof. By Proposition 4.12, νQis well defined for eacht≤0. By definition one has
t−1(ν(∂t+∇E+F +K)ν(Q)f) = (∂t+∇E+F+K)Q(t−1f) = 0.
The second equality follows by a similar argument.
One important observation from Theorem 4.13 is that the heat kernel of the vector representation is not a smoothing operator. However, if G ⇒M is a Lauter-Nistor groupoid in the sense of Definition 3.3, then for any f ∈Γ∞c (E|M0), one has
(23) ν(K)(f)(x) = Z a∈Gx κ(a−1)f(t(a))µx(a) = Z y∈Mα κ|GM0(x, y)f(y)µM0, where Mα is the connected component of M0 containing x and we have used the
4.1.4. Application: Heat kernel in edge calculus. As an application of our construction, we give a simple proof to Albin’s conjecture on generalization of [1, Theorem 4.3]. We refer to the same paper for details.
Theorem 4.14. A Laplacian operator on any manifolds M with iterated complete edge has a heat kernel.
Proof. By [3], the pseudo-differential calculus is defined by a groupoid G over the compactification M of M0. In particular, any Laplacian on M0 is the vector repre- sentation of a Laplacian operator on G. The lemma follows from our constructions
above.
4.2. Transverse regularity of the heat kernel. In the last Section, we proved that the series P∞
k=0(−1)kQ (k)
N (·, t) converges to the heat kernel Q(·, t) in the k · kl
norms. It follows that Q is smooth on eachs-fiber. In this section, we consider the problem of regularity of the heat kernel Q.
4.2.1. Riemannian metrics and connections on the groupoid G. Let G be a groupoid with compact units M, letsbe the source map. As in the beginning of this section, we have already fixed an invariant metric gA on the foliation ker(ds)⊂TG. We shall extend gA to TG. Fix a distribution H ⊂TG complementary to ker(ds). Then the differential ds identifies H ∼= s−1TM. It follows that any metric on M defines a metric gH n H. We define the metric gG on G by taking the orthogonal sum of Hand ker(ds).
The distribution H canonically induces a splitting
TG˜= ker(d˜s)⊕ker(d˜t)⊕ H(2),
such that H ={d˜s(X) : X ∈ H(2)} ={d˜t(X) : X ∈ H(2)} (see [13]). Indeed, one can write down H(2) explicitly:
H(2):= (H × H)\TG˜.
Also, note that the relation s(2) =s◦˜t=s◦˜s implies ker(ds(2)) = ker(d˜s)⊕ker(d˜t).
Given any metric gG as above, the splittingTG˜= ker(ds˜)⊕ker(d˜t)⊕ H(2) defines a metric on ˜G, which shall be denoted by ˜gG.
Next, we equip TG with a special connection, following [13]. Recall that one has identification TG = Ker(ds)⊕s−1T∗M. Denote the orthogonal projection onto Ker(ds) byPV. Take the Levi-Civita connection∇TGon (G, gG). Then∇TG induces a connection on V := Ker(ds) by
∇VXY :=PV∇TXGY ∀X∈TG, Y ∈Γ∞(Ker(ds))⊂Γ∞(TG).
We define the connection∇V⊕HonTG by taking the direct sum of∇V ands−1∇TM. 4.2.2. The regularity theorem. In this section, we state and prove our transverse regularity theorem. Let G ⇒ M be a groupoid with M compact. We shall assume that the Lie algebroidAis orientable. Letµbe thes-fiber-wise invariant Riemannian volume form.
Recall that ˜G:= {(a, b)∈ G × G :s(a) =s(b)} and me(a, b) =ab
−1, ∀(a, b)∈G˜. Also, we write me∗ to denote the differential of me, regarded as a bundle map, i.e.
e
m∈Γ∞(HomT( ˜G,me−1TG)); andL(m)µto denote them-th Lie derivative ofµ. (see Appendix A.1).
Theorem 4.15. Assume that
(1) The source maps:G →Mis a fiber bundle;
(2) For eachm∈N, there exist constantsCm, εm>0 such that
|(∇HomT( ˜G,me −1TG) )mme∗|(b 0, b)≤C meεm(ds(b 0,s(b0))+d s(b,s(b)));
(3) The Lie derivatives of µ satisfy the estimate |L(m)µ(X1H˜,· · · , XmH˜)(b0, b)| ≤Cmeεm(ds(b
0,s(b0))+d
s(b,s(b)))|X
1| · · · |Xm|.
Then for anyF ∈Γ∞(t−1E⊗s−1E0), K ∈Ψ−∞
µ (G,E), the heat kernele−t(∆
E+F+K)
∈
Γ∞b .
Proof. Recall from Lemma 4.9 that the heat kernel is defined to be the sum
e−t(∆E+F+K)= ∞
X
k=0
(−1)kQ(k),
where, using Equation (5), the Q(k) have the form:
Q(0)(a, t) =GN(a, t) Q(k)(a, t) = Z t 0 Z (b0,b)∈˜t−1(s(a)) e m−1GN(b0, b, t−τ)˜s−1R(k−1)(b0, b, τ)˜µ(˜b)dτ,
whereRN(k) is defined by taking convolution product ofR(1)N := (∂t+ ∆E+F+K)GN
with itself k-times.
Fix a connection∇Eon E→M. We denote by∇t−1E⊗s−1E0
to be the tensor of the pullbacks of ∇E by s and t. Hence∇t−1E⊗s−1E0 is a connection onG. Pulling-back again by ˜t, one has the bundle ˜t−1(t−1E⊗s−1E0) over ˜G, and the corresponding connection ∇˜t−1(t−1E⊗s−1E0)
.
We begin with estimating the covariant derivatives of RN(1). Taking covariant derivative throughout the proof of (2) of Lemma 4.6, one gets
∇t−1E⊗s−1E0(κ◦GN)→ ∇t
−1E⊗s−1E0
GN,
as t goes to 0. Modifying the arguments of the proof of (1) of Lemma 4.6 in the same manner, one gets the estimate
k(∇t−1E⊗s−1E0)m((∂t+ ∆E+F)GN)k0 ≤Cm(1)tN
−n 2−
l 2−m. Combining the two, it follows that
k(∇t−1E⊗s−1E0
)mR(1)N (·, t)k0≤Cm(1)(tN−n2+l−m+ 1) for some constants Cm(1) independent of t.
Next, we estimate the derivatives of R(Nk). Write
R(Nk)(a, t) = Z t 0 Z (b0,b)∈˜s−1(a) e m−1R(1)N (b0, b, t−τ)˜s−1RN(k−1)(b0, b, τ)˜µ(b0, b)dτ.
Then the corollaries of Lemma A.3 imply for any (local) vector field X on G, (∇Xt−1E⊗s−1E0R(Nk))(a, t) = Z t 0 Z (b0,b)∈˜s−1(a) ∇˜t(t−1E⊗s−1E0) me−1R(1)N (b0, b, t−τ)˜s−1R(Nk−1)(b0, b, τ)(XH˜)˜µ dτ + Z t 0 Z (b0,b)∈˜s−1(a) e m−1R(1)N (b0, b, t−τ)˜s−1RN(k−1)(b0, b, τ) L(1)µ˜(XH˜)dτ,
where XH˜ ∈ Γ( ˜H) ⊂ Γ(TG˜) is the horizontal lift of X. Observe that for all t > 0,
R(1)N (·, t) is supported on a set of the form{a∈ G :ds(a,s(a))≤ρ} for someρ >0.
It follows that R(Nk−1) is supported on the set {a∈ G :ds(a,s(a))≤(k−1)ρ}; and
e
m−1R(1)N (b0, b, t−τ) is supported on the compact set {ds(b0, b)≤ρ}.Hence, for each a∈ G fixed, the domain of integration can be re-written as
B(a, ρ) :={b∈ G :s(b) =s(a), ds(a, b)≤ρ},